RON AHARONI AND SHIRA ZERBIB A famous conjecture of Tuza ... · RON AHARONI AND SHIRA ZERBIB...

A GENERALIZATION OF TUZA’S CONJECTURE

RON AHARONI AND SHIRA ZERBIB

Abstract. A famous conjecture of Tuza [11] is that the mini-mal number of edges needed to cover all triangles in a graph isat most twice the maximal size of a set of edge-disjoint trian-gles. We propose a wider setting for this conjecture. For a hy-pergraph H let ν(m)(H) be the maximal size of a collection ofedges, no two of which share m or more vertices, and let τ (m)(H)be the minimal size of a collection C of sets of m vertices, suchthat every edge in H contains a set from C. We conjecture thatthe maximal ratio τ (m)(H)/ν(m)(H) is attained in hypergraphs forwhich ν(m)(H) = 1. This would imply, in particular, the follow-ing generalization of Tuza’s conjecture: if H is 3-uniform, thenτ (2)(H)/ν(2)(H) ≤ 2. (Tuza’s conjecture is the case in which His the set of all triples of vertices of triangles in any given graph.)We show that most known results on Tuza’s conjecture go over tothis more general setting. We also prove some general results onthe ratio τ (m)(H)/ν(m)(H), and study the fractional versions andthe case of k-partite hypergraphs.

1. Introduction

1.1. Terminology. A hypergraph is a pair H = (V = V (H), E =E(H)), where E is a collection of subsets of V , called edges. Theelements of V are called vertices. If all edges are of the same size kthen H is said to be k-uniform. A hypergraph is called linear if no twoedges intersect in more than one vertex.

We shall often identify the hypergraph with its edge set, and writeH for E(H). The degree degH(v) of a vertex v in a hypergraph H is thenumber of edges containing v. Let ∆(H) = maxv∈V (H) degH(v). Weshall use the abbreviation a1a2 . . . am to denote the set {a1, a2, . . . , am},and in this notation we assume that all ai are distinct. The notation(Sm

)stands for the set of all subsets of size m of the set S.

A set of vertices in a hypergraph H is called independent if it doesnot contain an edge. We denote by α(H) the maximal size of an inde-pendent set in H (this notation is used in the paper only for graphs).

The research of the first author was supported by BSF grant no. 2006099, by anISF grant and by the Discount Bank Chair at the Technion.

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2 RON AHARONI AND SHIRA ZERBIB

The line graph L(H) of a hypergraph H has E(H) as its vertices, andtwo vertices are connected if the corresponding edges intersect.

A matching in a hypergraph H is a set of disjoint edges. The match-ing number ν(H) is the maximal size of a matching in H. Clearlyν(H) = α(L(H). A cover is a set of vertices intersecting all edges ofH. The covering number τ(H) is the minimal size of a cover.

1.2. Tuza’s conjecture. Trivially, ν(H) ≤ τ(H) for any hypergraphH. The union of all edges in a maximal matching is a cover, and henceif H is k-uniform then

(1) τ(H) ≤ kν(H).

Equality is attained, for example, in k-uniform projective planes, whereν = 1 and τ = k, and in the complete k-uniform hypergraph

([(n+1)k−1]

k

),

namely, the collection of all k-subsets of a set of size (n + 1)k − 1,where ν = n and τ = kn. However, in special classes of hypergraphs,the inequality can be improved. For example, it is conjectured thatin k-partite hypergraphs τ ≤ (k − 1)ν (this conjecture is commonlyattributed to Ryser, and it appeared in a thesis of his student, Hen-derson [7]). Ryser’s conjecture is known for k = 3 [1], so in 3-uniform3-partite hypergraphs τ ≤ 2ν. A famous conjecture of Tuza is thatthe same inequality, τ ≤ 2ν, holds for the hypergraph whose verticesare the edges of a graph G, and its edges are the triangles in G, forany graph G. Below (Problem 1.8) we shall offer some indication thatthe similarity between Tuza’s conjecture and the case k = 3 of Ryser’sconjecture is not merely formal.

The main purpose of this article is to put Tuza’s conjecture in a gen-eral context. We shall formulate a more general conjecture (Conjecture1.2), that applies to all 3-uniform hypergraphs, and then extend thelatter to general uniformities (Conjecture 1.10). There are no new re-sults in the paper on Tuza’s conjecture itself, but the discussion of thegeneral case and of higher uniformities may shed light on the difficultiesit poses.

1.3. More notation. We say that a hypergraph is an m-matching ifno two edges in it intersect in m vertices or more. So, a 1-matching isplainly a matching. For a hypergraph H and a positive integer m letH(m) be the hypergraph whose vertex set is

(V (H)m

)and whose edge set

is {(em

)| e ∈ E(H)}. Clearly, a subset F of H is an m-matching if and

only if {(fm

)| f ∈ F} is a matching in H(m).

A cover of H(m) is said to be an m-cover of H. So, an m-cover of His a set C of m-sets of vertices, such that every edge in H contains at

A GENERALIZATION OF TUZA’S CONJECTURE 3

least one member of C. We write ν(m)(H) = ν(H(m)) and τ (m)(H) =τ(H(m)), namely, ν(m)(H) and τ (m)(H) are the maximal size of an m-matching and the minimal size of an m-cover in H, respectively.

For m < k let H(k,m) = {H(m) | H is a k-uniform hypergraph}.Define

g(k,m) = max{τ(J) | J ∈ H(k,m), ν(J) = 1}and

h(k,m) = sup{τ(J)

ν(J)| J ∈ H(k,m)

}.

Clearly, g(k,m) ≤ h(k,m).We shall also consider the fractional versions of these parameters.

Let H = (V,E) be a hypergraph. A function f : E → R≥0 is afractional matching in H if

∑e3v f(e) ≤ 1 for every v ∈ V . The

fractional matching number ν∗(H) is the maximum of∑

e∈E f(e) overall fractional matchings f . A function g : V → R≥0 is a fractional coverin H if

∑v∈e g(v) ≥ 1 for every e ∈ E. The fractional covering number

τ ∗(H) is the minimum of∑

v∈V g(v) over all fractional covers g. Linearprogramming duality implies ν(H) ≤ ν∗(H) = τ ∗(H) ≤ τ(H). Wedefine

g∗(k,m) = sup{τ ∗(J) | J ∈ H(k,m), ν(J) = 1}

h∗(k,m) = sup{τ ∗(J)

ν(J)| J ∈ H(k,m)

},

and

j∗(k,m) = sup{ τ(J)

ν∗(J)| J ∈ H(k,m)

}.

Let ν∗(m)(H) = ν∗(H(m)) and τ ∗(m)(H) = τ ∗(H(m)).

1.4. A general conjecture. For a graph G let T (G) be the 3-uniformhypergraph whose vertex set is V (G), and whose edges are all triplesof vertices forming triangles in G. Tuza’s conjecture can be written as:

Conjecture 1.1. [11] If H = T (G) for a graph G, then τ (2)(H) ≤2ν(2)(H).

It may well be the case that this is true for all 3-uniform hypergraphs.

Conjecture 1.2. For every 3-uniform hypergraph H, τ (2)(H) ≤ 2ν(2)(H).

Since there are simple examples showing h(3, 2) ≥ 2, the conjecturesays that h(3, 2) = 2.

The 2-skeleton of a hypergraph H is the graph obtained by con-necting by an edge every pair of vertices {u, v} that is contained insome edge of H. A hypergraph is 2-connected if its 2-skeleton graph is2-connected.


Tuza’s conjecture has only two known 2-connected examples showingsharpness, G = K4 and G = K5. Conjecture 1.2, by contrast, has aninfinite family of 2-connected instances in which equality is attained.

Example 1.3. For n even, let H be the hypergraph whose edges are allsubsets of size 3 of [n] containing 1. Then ν(2)(H) = ν(Kn−1) = n−2

2(an

example of a 2-matching of this size is {(1, 2i, 2i + 1) | 1 ≤ i ≤ n−22}),

while τ (2)(H) = τ(Kn−1) = n − 2 (a minimal cover is {(1, i) | 1 < i <n}).

This example shows that the class of triangle hypergraphs T (G) isa proper subset of H(3, 2), and that Conjecture 1.2 is a proper ex-tension of Tuza’s conjecture. Apart from this family, all examples of2-connected hypergraphs showing tightness that we know are small:T (K4) and T (K5),

([5]3

)minus one triple, and

([5]3

)minus two triples

that intersect in one vertex. In the last two examples ν(2) = 2 andτ (2) = 4. The hypergraph

([4]3

)minus one triple is also an example of

tightness, but it falls within the scope of Example 1.3.Conjecture 1.2 can be generalized to hypergraphs of all uniformities.

Conjecture 1.4. h(k,m) = g(k,m) for all m ≤ k.

Conjecture 1.2 would follow from this and the fact proved below(Proposition 3.7), that g(3, 2) = 2.

Example 1.3 can also be generalized. Let m < k. For any (k − 1)-uniform hypergraph L let x be a vertex not belonging to V (L), and letL = {{x}∪e | ` ∈ L}. Then τ (m)(L) = τ (m−1)(L), ν(m)(L) = ν(m−1)(L)and τ ∗(m)(L) = τ ∗(m−1)(L). This observation implies:

Proposition 1.5. g(k,m) ≥ g(k−1,m−1), h(k,m) ≥ h(k−1,m−1),h∗(k,m) ≥ h∗(k − 1,m− 1), and j∗(k,m) ≥ j∗(k − 1,m− 1).

In particular

(2) h(k, 2) ≥ h(k − 1, 1) = k − 1

As we shall later see, this inequality is in some cases strict, e.g.,h(4, 2) ≥ 4.

Conjecture 1.2 indicates thatH(3, 2) hypergraphs behave better thangeneral 3-uniform hypergraphs. Indeed, H(3, 2) is a narrow subclass ofthe class of 3-uniform hypergraphs. For example, it is easy to check thatH(3, 2) hypergraphs are linear. It is not clear whether there is a finitelist of forbidden substructures that characterizes H(3, 2) hypergraphs.But it is possible that a milder condition than being in H(3, 2) sufficesto guarantee τ(H) ≤ 2ν(H).


Figure 1. A 3-unfiorm linear tent.

Definition 1.6. A tent in a hypergraph H is a set of four edges {ei, 1 ≤i ≤ 4} satisfying:

(1)⋂3i=1 ei 6= ∅,

(2) |e4 ∩ ei| = 1 for all i ∈ {1, 2, 3}, and(3) e4 ∩ ei 6= e4 ∩ ej for i 6= j.

Note that ifH is linear then a tent is a set of four pairwise intersectingedges e, f, g, h, such that e ∩ f ∩ g = {x} and x /∈ h (see Figure 1).

Lemma 1.7. If H is k-uniform then H(k−1) does not contain a tent.

Proof. Clearly, H(k−1) is linear. Assume for contradiction that it con-tains a tent e, f, g, h as above. Let e′, f ′, g′, h′ be edges in H such thate =

(e′

k−1

), f =

(f ′

k−1

), g =

(g′

k−1

), h =

(h′

k−1

). Let and x = {v1, . . . , vk−1}

be the intersection vertex of e, f, g. Then e′ = {v1, . . . , vk−1, u1}, f ′ ={v1, . . . , vk−1, u2},g′ = {v1, . . . , vk−1, u3}, where vi, uj are all distinct vertices of H. Sinceh′ intersects each of e, f, g at a vertex different than x, h′ must con-tain u1, u2, u3 and at least k − 2 vertices from v1, . . . , vk−1, implying|h′| ≥ k + 1, a contradiction. �

It is easy to realize that also 3-partite 3-uniform hypergraphs cannotcontain tents. Hence it is tempting (though too daring to state as a con-jecture) to pose the following problem, concerning a possible commongeneralization of Conjecture 1.2 and the 3-case of Ryser’s conjecture:

Problem 1.8. Is it true that a 3-uniform hypergraph not containinga tent satisfies τ ≤ 2ν?


A special family of hypergraphs that do not contain a tent is thefamily of hypergraphs H with ∆(H) ≤ 2. For this family the answeris positive, a fact that is generalized in the following.

Theorem 1.9. If H is an k-uniform hypergraph with ∆(H) ≤ 2, thenτ(H) ≤ dk+1

2eν(H).

The dual D of Kk+1 is an example for sharpness of the theorem. Thevertices of D are the edges of Kk+1, and to each vertex v of Kk+1 therecorresponds the edge ev = {e ∈ E(Kk+1) | v ∈ e} of D (the star ofv). Then ∆(H) = 2 because every vertex, namely edge uv of Kk+1,belongs to two stars, eu and ev. Also, τ(D) = dk+1

2e and ν(D) = 1.

We shall also see later that some of the results related to Conjecture1.2 use only the condition of not containing a tent (see Proposition 2.4and Proposition 3.7). It will be of interest to deduce still further resultsfrom this condition.

1.5. Aims, structure of the paper, and main results. Determin-ing precise values of h(k,m), and even of g(k,m), seems difficult, asreflected by the fact that even the first non-trivial value, h(3, 2), isunknown. Besides generalizing Tuza’s conjecture, the main aim of thepaper is obtaining bounds on these functions and on their fractionalversions.

In Section 2 we recapitulate some of the results known on Tuza’sconjecture, whose proofs go over to the same results on Conjecture 1.2.In particular we mention, without proof, the following results whoserestrictions to the setting of Tuza’s conjecture are known:

h(3, 2) ≤ 3− 3

23,

andj∗(3, 2) = 2.

We also prove that h∗(k, k − 1) ≤ k − 1, of which the case k = 3 isproved in [8].

In Section 3 we determine values of the above functions for someparameters, and find bounds on these functions for other parameters.In particular we show:

g∗(k, 2) =k2

4+ o(k2),

g(k, k − 1) =⌈k + 1

2

⌉,

g(4, 2) = 4,


2.5 ≤ h∗(4, 2) ≤ 4.5,

andj∗(4, 2) ≤ 4.

If Conjecture 1.4 is true, then the second result above will imply ageneralization of Conjecture 1.2 to all uniformities.

Conjecture 1.10. For every k-uniform hypergraph H, τ (k−1)(H) ≤⌈k+12

⌉ν(k−1)(H).

In Section 5 we discuss the case of k-partite hypergraphs. If H is3-partite, then so is H(2), and hence, by a result mentioned above,τ (2)(H)/ν(2)(H) ≤ 2. We show that if H is a 3-partite hypergraphthen τ ∗(2)(H)/ν(2)(H) ≤ 9/5, and if one of the vertex classes of His of size at most 2 then τ (2)(H)/ν(2)(H) ≤ 5/3. These results areprobably far from being sharp: we do not know any example of a 3-partite hypergraph H for which τ (2)(H)/ν(2)(H) > 4/3. We also provebounds on τ ∗(k−1)(H)/ν(k−1)(H) for every k-partite hypergraph H.

2. Known bounds on Tuza’s conjecture that go over tothe setting of Conjecture 1.2

The best bound known on the ratio τ (2)(T (G))/ν(2)(T (G)) for allgraphs G was proved by Haxell [5]. The proof applies, almost verbatim,to the general setting of Conjecture 1.2. Namely:

Theorem 2.1. For every 3-uniform hypergraph H, τ (2)(H) ≤ 6623ν(2)(H).

In other words, h(3, 2) ≤ 6623

.

Krivelevich [8] proved two fractional versions of Tuza’s conjecture.

Theorem 2.2. [8] The following are true for every graph G:

(1) τ (2)(T (G)) < 2ν∗(2)(T (G)).(2) τ ∗(2)(T (G)) ≤ 2ν(2)(T (G)).

Part (1) is true, with the same proof, for all 3-uniform hypergraphs,namely:

Theorem 2.3. For every 3-uniform hypergraph H, τ (2)(H) < 2ν∗(2)(H).

This cannot be improved in general, as is shown by the hypergraphsin Example 1.3. They satisfy ν∗(2)(H) = ν∗(Kn−1) = n−1

2, and thus

the ratio τ (2)(H)

ν∗(2)(H)= 2(n−2)

n−1 approaches 2 when n→∞. Combined with

Theorem 2.4, this entails

j∗(3, 2) = 2.


The generalization of the second part of Theorem 2.2 follows fromLemma 1.7.

Proposition 2.4. If k ≥ 3, then h∗(k, k − 1) ≤ k − 1.

Proof. By a theorem of Furedi [4], a k-uniform hypergraph not contain-ing a copy of a projective plane of uniformity k satisfies τ ∗ ≤ (k− 1)ν.Since for k ≥ 3 a k-uniform projective plane contains a tent, the resultfollows by Lemma 1.7. �

Note that Proposition 2.4 is not true for k = 2, since in this caseH(k, k − 1) is the set of all graphs.

3. Some values and some bounds

Proposition 3.1. The supremum in the definition of g∗(k,m) is at-tained.

Proof. It suffices to show that for every number t there are essentiallyfinitely many hypergraphs in H(k,m) with τ ∗ ≥ t. Put rigorously,this means that there exists p = p(k,m) such that if H is a k-uniformhypergraph with more than p edges and ν(m)(H) = 1 then there exists ak-uniform hypergraph K with ν(m)(K) = 1, |K| < |H|, and τ ∗(m)(K) ≥τ ∗(m)(H).

By the Sunflower Lemma [3] there exists p such that if |H| > pthen there exists a set of edges S ⊆ E(H) having k + 1 edges, suchthat all pairwise intersections of edges in S are the same set C. Everyf ∈ E(H) is disjoint from at least one of the (disjoint) sets e\C, e ∈ S,and since the fact that ν(m)(H) = 1 implies that f intersects every edgein H in at least m vertices, we have |f ∩C| ≥ m, and thus in particular|C| ≥ m.

Let K be obtained from H by replacing all edges of S by the singleedge C∪T , where T is a set of size k−|C|, disjoint from V (H). To finishthe proof it suffices to show that τ ∗(m)(K) ≥ τ ∗(m)(H), namely that for

every fractional m-cover g :(V (K)m

)→ R+ of K there is a fractional

m-cover of H of equal or smaller size. For every set a ∈(V (K)m

)not

contained in V (H) (namely containing vertices from T ) replace g(a)by the same weight on b, where b is a set of size m containing a ∩ Cand contained in C. Note that the same b may aggregate weights fromdifferent sets a. The resulting function is a fractional m-cover in H,with size at most g. �

3.1. Bounds on g(k, 2) and on g∗(k,m). If H is k-uniform andν(2)(H) = 1, then for any edge h ∈ H the set

(h2

)is a 2-cover for

H. Hence g(k, 2) ≤(k2

). This bound is not attained for any k > 2.


Proposition 3.2. If k > 2 then g(k, 2) <(k2

).

Proof. Let H be a k-uniform hypergraph, k ≥ 3, with ν(2)(H) = 1.Write

r = max{|e ∩ f | | e, f ∈ E(H)},and let t = min{r, dk/2e}. Then t ≥ 2. We show that τ (2)(H) ≤(k2

)−(t2

).

Indeed, let e, f be two edges in H with |e ∩ f | ≥ t. Write

e = {v1, v2, . . . , vt, ut+1, . . . , uk},

f = {v1, v2, . . . , vt, wt+1, . . . , wk}.Let Ae and Af be the first dk/2e vertices of e and f in the above order,respectively. Thus {v1, v2, . . . , vt} ⊂ Ae ∩ Af . Let Be = e \ Ae andBf = f \ Af . Then the set(

Ae2

)∪(Af2

)∪ (Be ×Bf )

is a 2-cover of H of size at most(k2

)−(t2

). �

Calculating g(k, 2) for general k is probably hard, but we know thatit is of quadratic order. The upper bound, g(k) <

(k2

), was noted

above. The lower bound follows from the next theorem, and the factthat g(k,m) ≥ g∗(k,m).

Theorem 3.3. g∗(k, 2) = k2

4+ o(k2).

Proof. By the Prime Number Theorem, for k large there exists a primep < k

2− 1 such that k

2− p = o(k). Let r = p + 1 and let P be a

projective plane of uniformity r. Let K be the union of two disjointcopies P1, P2 of P , namely

K = {e1 ∪ e2 | e1 ∈ E(P1), e2 ∈ E(P2)}.Then K is 2r-uniform and ν(2)(K) = 1. Every pair a, b of distinctvertices ofK is contained in no more than r2 edges ofK, so the constantfunction assigning value 1

r2to every edge of K is a fractional 2-matching

of K of size (r2−r+1)2

r2. This proves that g∗(2r, 2) ≥ (r2−r+1)2

r2. This

implies that g∗(k, 2) ≥ (r2−r+1)2

r2= k2

4+ o(k2).

For the other direction, we prove that for every k, g∗(k, 2) ≤ k2

4+k−2.

Let H be a k-uniform hypergraph with ν(2)(H) = 1. If every two edgesin H meet at 3 or more vertices, then the function assigning 1

3to every

pair of vertices contained in some fixed edge e ∈ H, and 0 to everyother pair of vertices, is a fractional 2-cover of size k(k − 1)/6.


Otherwise, there exist two edges e, f ∈ H meeting at exactly two ver-tices. Write e = {w1, w2, v1, . . . , vk−2} and f = {w1, w2, u1, . . . , uk−2},where the vertices vi and uj are all distinct. Since ν(2)(H) = 1, bothe and f share at least two vertices with every edge in H. This impliesthat the function c defined below is a fractional 2-cover for H:

c(w1w2) = 1

c(wivj) = 1, 1 ≤ i ≤ 2, 1 ≤ j ≤ k − 2

c(viuj) = 14, 1 ≤ i ≤ k − 2, 1 ≤ j ≤ k − 2

The size of c is (k−2)24

+ 2k−3 = k2

4+k−2, which yields the result. �

Next, we note that the example in the proof of Theorem 3.3 can beextended to m > 2:

Example 3.4. Let r be such that there exists a projective plane P ofuniformity r. Let P1, . . . , Pm be m disjoint copies of P , and considertheir join,

H = P ∗m = {e1 ∪ · · · ∪ em | ei ∈ E(Pi), 1 ≤ i ≤ m}.Then H is a k-uniform hypergraph for k = mr, and ν(m)(H) = 1.Moreover, for a1, . . . , am ∈ V (H) we have deg(a1a2 . . . am) ≤ rm inH(m), where the maximum is attained when ai ∈ V (Pi), 1 ≤ i ≤ m.Hence the constant function f : E(H) → R+ assigning 1

rmto every

edge is a fractional m-matching, of size

(r2 − r + 1)m

rm≥ km

mm− o(km).

Therefore, by the same argument as above, we have:

Proposition 3.5. For every 2 ≤ m < k − 1, g∗(k,m) ≥ km

mm − o(km).

3.2. The function g(k, k − 1). We wish to prove that g(k, k − 1) =dk+1

2e. This will follow from Theorem 1.9, which we prove below. The

proof uses the following lemma.

Lemma 3.6. Let H be a k uniform hypergraph satisfyin ∆(H) ≤ 2,and let G = L(H). If G is a complete graph or an odd cycle thenτ(H) ≤ dk+1

2e.

Proof. Suppose first that G is complete, meaning that ν(H) = 1. Since∆(H) ≤ 2, every edge meets at most k other edges, meaning that|E(H)| ≤ k + 1. It follows that τ ≤ dk+1

2e (choose a vertex at a time

belonging to the intersection of two edges, and remove these edges).If G is a cycle of length 2r + 1 for some r ≥ 1, then ν(H) = r and

τ(H) = r + 1. Assuming (as we may do) that k > 1, it follows thatτ(H)ν(H)

= r+1r≤ dk+1

2e. �


Proof of Theorem 1.9. It clearly suffices to prove the theorem forconnected hypergraphs. Let m = |E(H)|, and let G = L(H) be theline graph of H. Then ν(H) = α(G). Let M be an inclusion-wisemaximal matching in G, and let p = |M |. By the maximality of M ,the set V (G) \

⋃M is independent, and thus α(G) ≥ m− 2p, or

p ≥ m− α(G)

2.

Each pair {e, f} of edges of H represented by an edge in M can becovered in H by a vertex in e ∩ f , and each edge of H representedby a vertex in V (G) \

⋃M can be covered by a single vertex. Hence

τ(H) ≤ p + (m − 2p) = m − p. On the other hand, the fact that∆(H) ≤ 2 implies that ∆(G) ≤ r.

By the lemma, we may assume that G is neither a complete graphnor an odd cycle, and since we are assuming that G is connected, byBrooks’ theorem [2], α(G) ≥ m

r. Summarizing, we have:

τ(H) ≤ m− p ≤ m+ α(G)

2≤ α(G)(r + 1)

2=ν(H)(r + 1)

2,

proving the theorem. �

Proposition 3.7. g(k, k − 1) = dk+12e.

Proof. Let show first that g(k, k − 1) ≤ dk+12e, namely τ(H(k−1)) ≤

dk+12eν(H(k−1)) for every k-uniform hypergraph H with ν(H(k−1)) = 1.

If ∆(H) ≤ 2, then the inequality follows from Theorem 1.9. So, we mayassume that there exists v ∈ V (H(k−1)) having degree greater than 2.By Lemma 1.7 H(k−1) does not contain a tent, which, since all edgesin H(k−1) intersect, means that there is no edge missing v, implyingτ(H(k−1)) = 1.

For the converse inequality consider the hypergraph H =([k+1]k

).

Clearly, ν(k−1)(H) = 1. For a (k − 1)-set t ⊂ [k + 1], the complementtc = [k+ 1] \ t is a pair, and a set of (k− 1)-tuples ti is a (k− 1)-coverfor H if the corresponding pairs tci cover all vertices of [k + 1]. Henceτ (k−1)(H) = dk+1

2e. �

3.3. Some values of the functions, and some bounds.

Proposition 3.8. g(4, 2) = 4.

Proof. Let H be the hypergraph on vertex set {a, b, c, d, e, f, g} withedge set {abcd, abef, cdef, aceg, bdeg, adfg, bcfg}. Then ν(2)(H) = 1and τ (2)(H) = 4. This shows g(4, 2) ≥ 4.


To prove the converse inequality, let H be a 4-uniform hypergraphwith ν(2)(H) = 1. We show that τ (2)(H) ≤ 4. Let e be an edge in H.Call a pair a ∈

(e2

)indispensable if there is an edge f ∈ E(H) such

that f ∩ e = a, and otherwise call it dispensable. Note that if an edgef 6= e contains two pairs from

(e2

), then it contains also a third pair.

Hence, if a, b ∈(e2

)are dispensable pairs, then

(e2

)\ {a, b} is a 2-cover

for H of size 4. Hence we may assume that there exists at most onedispensable pair. So, there exist indispensable pairs a, a′, b, b′ such thata ∩ a′ = b ∩ b′ = ∅.

Let f, f ′, g, g′ be edges in H witnessing the indispensability of thepairs a, a′, b, b′, respectively, namely, f ∩ e = a, f ′ ∩ e = a′, g ∩ e =b, g′∩e = b′. Since ν(2)(H) = 1, f∩f ′ is a pair x disjoint from e. Thereis no other edge intersecting e just at a, since such an edge would notintersect f ′ in two vertices or more. Similarly, also f ′, g, g′ are the onlywitnesses to the indispensability of a′, b, b′ respectively. Let y = g ∩ g′.By the above argument, the set of pairs

(e2

)∪ {x, y} \ {a, a′, b′, b′} is a

2-cover for H of size 4, proving the theorem. �

Proposition 3.9. 2.5 ≤ h∗(4, 2) ≤ 4.5.

Proof. The lower bound is valid since g∗(4, 2) ≥ 2.5, as shown by the

hypergraph H =([6]4

), for which ν(2)(H) = 1 and τ ∗(2)(H) = 2.5 (an

optimal fractional 2-matching gives each of the 15 edges weight 1/6,and an optimal fractional 2-cover gives each of the 15 pairs of verticesweight 1/6).

To prove the upper bound, let H be a 4-uniform hypergraph, andlet ν(2)(H) = ν. We shall show that τ ∗(2)(H) ≤ 4.5ν. We constructa fractional 2-cover of H as follows. Let M = {m1, . . . ,mν} be a 2-matching in H. First, assign weight 1/2 to every pair in P (M) =⋃νi=1

(mi

2

). This amounts to a total weight of 3k, and every edge in

H that intersects each edge in M in at least two pairs is fractionally2-covered.

For every 1 ≤ i ≤ ν and every pair p ∈(mi

2

), let q(p) = mi\q, and let

H(p) be the set of edges e ∈ H for which e ∩mi = p, and |e ∩mj| < 2for all i 6= j. Note that

⋃νi=1

⋃p∈(mi

2 )H(p) is the set of those edges that

are not yet fractionally 2-covered.If for some p ∈

(mi

2

)we have e ∈ H(p) and f ∈ H(q(p)), then

|e ∩ f \mi| = 2, since otherwise M ∪ {e, f} \ {mi} is a 2-matching ofsize ν+1. This means that there exists a pair r(p) of vertices containedin all edges belonging to H(p) ∪ H(q(p)). In particular, if H(p) = ∅then r(p) = q(p). We assign now additional weights of 1

2to r(p), for

every p ∈ P (M) (so, if r(p) = q(p) then it now has weight 1). We have


thus added weight 12

for every one of the three pairs of disjoint vertex

couples in(mi

2

), so in total we added 3k

2to the fractional 2-cover we

construct. Now every edge in H is fractionally 2-covered, and the totalassigned weight is at most 4.5ν. �

Proposition 3.10. j∗(4, 2) ≤ 4.

Proof. We shall show that ifH is a 4-uniform hypergraph then τ (2)(H) <4τ ∗(2)(H). Assume, for contradiction, that there exists a 4-uniform hy-pergraph H for which τ (2)(H) ≥ 4τ ∗(2)(H), and let H be minimal withrespect to this property.

Let g, f be a minimal fractional 2-cover and maximal fractional 2-matching in H, respectively. Let U be the set of all pairs u ∈

(V (H)

2

)for which g(u) > 0. By a complementary slackness condition,

(3) |U | =∑u∈U

∑e∈H(2):u∈e

f(e) ≤ 6ν∗(2)(H).

If there exists u ∈ U with g(u) ≥ 1/4, let H ′ be the hypergraphobtained from H by removing all edges in H containing u. Thenτ (2)(H) ≤ τ (2)(H ′) + 1 and τ ∗(2)(H) ≥ τ ∗(2)(H ′) + 1

4. By the mini-

mality assumption on H we have τ (2)(H ′) < 4τ ∗(2)(H ′), implying that

τ (2)(H) ≤ τ (2)(H ′) + 1 < 4τ ∗(2)(H ′) + 1

≤ 4(τ ∗(2)(H)− 1/4) + 1 = 4τ ∗(2)(H),

contradicting the assumption on H.Thus we may assume that g(u) < 1/4 for all u ∈ U , implying that

every edge h ∈ H contains at least 5 members of U . Considering U as agraph on V (H), there is a partition of V (H) into sets A,B such that atleast |U |/2 edges in U are (A,B)-crossing, namely, have a non-emptyintersection with both A and B. The set(

U ∩(A

2

))∪(U ∩

(B

2

))forms a 2-cover for H, since in every edge in H there can be only 4(A,B)-crossing edges from U , and as noted above every edge in Hcontains 5 pairs belonging to U . Thus we have τ (2)(H) ≤ |U |/2, which,together with (3) yields

τ (2)(H) ≤ |U |/2 ≤ 3τ ∗(2)(H) < 4τ ∗(2)(H),

again contradicting the assumption on H. �


4. k-partite hypergraphs

A k-uniform hypergraph H = (V,E) is called k-partite if V is thedisjoint union of sets V1, . . . , Vk (called vertex classes), and every edgein E intersects every Vi at one vertex. If H is k-partite then H(m) is(km

)-partite, with vertex classes indexed by the sets A ∈

([k]m

). For each

e ∈ E(H) the edge(em

)of H(m) consists of one m-tuple in each class

indexed by a set A, namely e ∩⋃i∈A Vi.

k-partite hypergraphs behave particularly well with respect to match-ings. The best-known result in this direction is Konig’s theorem, stat-ing that in bipartite graphs τ = ν. Lovasz [9] proved that in k-partitehypergraphs τ < k

2ν∗, and as already noted, by a theorem of Furedi [4]

in such hypergraphs τ ∗ ≤ (k − 1)ν.As mentioned above, the k = 3 case of Ryser’s conjecture, namely

τ ≤ 2ν in 3-partite hypergraphs, is known. If H is 3-partite then so isH(2), and thus τ (2)(H) ≤ 2ν(2)(H). In [6] this was strengthened to:

Theorem 4.1. If G is a 3-partite graph then τ (2)(T (G)) < 1.956ν(2)(T (G)).

Careful scrutiny shows that the proof also yields the more general

Theorem 4.2. If H is a 3-partite hypergraph then τ (2)(H) < 1.956ν(2)(H).

In the following example, H is a 3-partite hypergraph with ν(2)(H) =3 and τ (2)(H) = 4.

Example 4.3. Let V = {a1, a2}∪ {b1, b2, b3}∪ {c1, c2} and let E consistof the edges a1b1c1, a1b1c2, a1b2c2, a2b2c2, a2b2c1, a2b3c1, a1b3c1.

We do not know any 3-partite hypergraph H in which τ (2)(H)

ν(2)(H)> 4

3. It

is easy to see that if H is 3-partite and ν(2)(H) = 1 then τ (2)(H) = 1,so this example shows that the analogue of Conjecture 1.4 is false inthe 3-partite case.

We shall be able to prove some bounds smaller than 2 on the ratioτ (2)(H)

ν(2)(H)for a 3-partite hypergraph H under two special conditions on

H: having one vertex class of size 2, or having identical neighborhoodsfor all vertices in one vertex class.

Let H be a 3-partite graph, with vertex classes A,B,C. The vertexclasses of H(2) are AB,BC and AC, where XY = {xy | x ∈ X, y ∈ Y }for any two sets X, Y . We introduce an asymmetry, by singling out onevertex class of H, say A. Write A = {a1, . . . , ap}, and for each i ≤ plet Fi = {bc | aibc ∈ H} and F = (F1, . . . , Fp). Then Fi are sets ofedges in a bipartite graph. Given such a family F , let ν(2)(F) be themaximum of |

⋃i≤pNi|, where each Ni is a matching in Fi. Let also

τ (2)(F) = minZ⊆B×C |Z|+∑

i≤p τ(Fi − Z).


Claim 4.4.

(1) ν(2)(F) = ν(2)(H).(2) τ (2)(F) = τ (2)(H).

Proof. (1) Given a 2-matching M in H, let Ni = {bc | aibc ∈M}. Theneach Ni is a matching in Fi, and M =

⋃i≤pNi. Conversely, if Ni, i ≤ p

are matchings in the respective sets Fi then⋃i≤p{aibc | bc ∈ Ni} is a

2-matching in H of size |⋃Ni|.

(2) Given a 2-cover Q of H, let Z = Q ∩ (B × C), and let Ti ={x ∈ B ∪ C | aix ∈ Q}. Then, for i ≤ p, Ti is a cover for Fi − Z,and |Q| = |Z| +

∑i≤p |Ti|. This proves that τ (2)(H) ≤ τ (2)(F). The

converse inequality is proved in a similar way. �

By Theorem 4.2 we have τ (2)(F) < 2ν(2)(F). If |F| = 1 then byKonig’s duality theorem τ (2)(F) = ν(2)(F), meaning that if |A| =1 then τ (2)(H) = ν(2)(H). Another case of equality is given in thefollowing:

Proposition 4.5. If Fi = Fj for all 1 ≤ i < j ≤ p then ν(2)(F) =τ (2)(F).

Recall that a set K of edges in a graph is a p-factor if ∆(K) :=maxv∈V degK(v) ≤ p. Let νp(G) be the maximal size of a p-factor ina graph G. By Konig’s edge coloring theorem, if G is bipartite then∆(K) ≤ p if and only if K is the union of p matchings. Letting G = Fi,we see that the following is a re-formulation of Proposition 4.5.

Proposition 4.6. If G is bipartite then νp(G) = minZ⊆E(G) |Z| +pτ(G− Z).

Proof. To show that νp(G) ≤ |Z| + pτ(G − Z) for every subset Z ofE(G), let C be a cover for G − Z, and consider a p-factor F . Sinceevery vertex in C is incident with at most p edges in F , We have

|F | = |F ∩ Z|+ |F \ Z| ≤ |Z|+ p|C|.

To prove the converse inequality, suppose that the respective vertexclasses of G are B,C, and let M be the matroid on E(G) consistingof those sets of edges F such that degF (b) ≤ p for every b ∈ B, andlet N be the matroid consisting of those sets of edges F such thatdegF (c) ≤ p for every c ∈ C. Note that for a set K of edges

(4) rankM(K) =∑b∈B

min(degK(b), p),

and similarly for N .


A set of edges is a p-factor if and only if it belongs to M ∩ N ,and hence, by Edmonds’ two matroids intersection theorem, νp(G) isthe minimum, over all partitions (E1, E2) of E(G), of rankM(E1) +rankN (E2).

Let (E1, E2) be a partition in which this minimum is attained. SetT1 = {bc ∈ E1 | degE1

(b) ≥ p} and T2 = {bc ∈ E2 | degE2(c) ≥ p}, and

let Z = E(G) \ (T1 ∪ T2). Then the set

{b ∈ B | degK(b) ≥ p} ∪ {c ∈ C | degK(c) ≥ p}is a cover for G− Z, and hence, by (4), we have

rankM(E1) + rankN (E2) ≥ |Z|+ kτ(G− Z),

proving that νp(G) ≥ |Z|+ pτ(G− Z). �

Corollary 4.7. If the neighborhoods of all vertices in one vertex classof a 3-partite hypergraph H are identical, then ν(2)(H) = τ (2)(H).

Another case in which we can improve the upper bound on the ratioτ (2)/ν(2) is that of |F| = 2.

Theorem 4.8. For any two sets F1, F2 of edges in a bipartite graph

τ (2)(F1, F2) ≤5

3ν(2)(F1, F2).

Proof. Let N be a maximum matching in F1∩F2, and let n = |N |. LetLi be a maximum matching in Fi \N , and let `i = |Li| (i = 1, 2). Bythe maximality of N we have |L1 ∩ L2| ≤ n, implying

|L1 ∪ L2| ≥ `1 + `2 − n.Assume without loss of generality that `1 ≥ `2. By the definition of

ν(2)(F ) we have

ν(2)(F ) ≥ max(n+ `1, |L1 ∪ L2|) ≥ max(n+ `1, `1 + `2 − n).

On the other hand, taking Z = N in the definition of τ (2)(F ) yields

τ (2)(F ) ≤ n+ `1 + `2.

Write `i = (1 + αi)n (where αi may be negative). By the above, inorder to prove the theorem it suffices to show that

3 + α1 + α2 ≤5

3max(2 + α1, α1 + α2 + 1).

If α2 ≤ 1 then the inequality is 3 + α1 + α2 ≤ 53(2 + α1), which after

cancellations becomes α2 ≤ 23α1 + 1

3, which is true since α2 ≤ α1 and

α2 ≤ 1. If α2 ≥ 1 the inequality is 3 + α1 + α2 ≤ 53(1 + α1 + α2). This

is valid since α2 ≤ α1, so α1 + α2 ≥ 2. �


Combined with Claim 4.4 this yields:

Corollary 4.9. If H is a 3-partite hypergraph with a vertex class ofsize 2, then τ (2)(H) ≤ 5

3ν(2)(H).

We conclude with a bound on the ratio between the fractional (k−1)-covering number and the integral (k−1)-matching number in k-partitehypergraphs.

Theorem 4.10. If H is a k-partite hypergraph then

τ ∗(k−1)(H) ≤ k2

2k − 1ν(k−1)(H).

In particular, a 3-partite hypergraph H has τ ∗(2)(H) ≤ 1.8ν(2)(H).

To prove the theorem, let H be a k-partite hypergraph with n =ν(k−1)(H), and let M be a (k − 1)-matching in H of size n. An edgee ∈ E(H) \M is said to mimic an edge m ∈ M if (M \ {m}) ∪ {e}is also a (k − 1)-matching (of size n) in H. By the maximality of M ,M ∪ {e} is not a (k − 1)-matching, and hence e mimics m if and onlyif |m ∩ e| = k − 1, and for every edge m′ ∈ M , m′ 6= m, we have|m′∩e| < k−1. Let M1 = {m1, . . . ,mt} be the set of those edges in Mthat have a mimicking edge, and let M2 = {mt+1, . . . ,mn} be the setof those edges that do not have a mimicking edge. For every 1 ≤ i ≤ tlet Fi ⊆ E(H) \ M be the set of all edges mimicking mi. If F is ahypergraph, we denote by

⋂F the intersection of its edges.

Claim 4.11. For every 1 ≤ i ≤ t, |⋂Fi ∩mi| = k − 1.

Proof. Suppose that this is false. Then there exist edges f, g ∈ Fi suchthat f ∩ mi 6= g ∩ mi. Since |f ∩ mi| = |g ∩ mi| = k − 1, and H isk-partite, this implies that |f ∩g| ≤ k−2. Therefore, M ∪{f, g}\{mi}is a (k − 1)-matching of size n+ 1 in H, a contradiction. �

For every 1 ≤ i ≤ t let pi =⋂Fi ∩mi.

Claim 4.12. The function g1 :(V (H)k−1

)→ R+, in which every pi, 1 ≤

i ≤ t, is mapped to 1, and every other set of k − 1 vertices containedin mi, 1 ≤ i ≤ n, is mapped to 1/2, is a fractional (k − 1)-cover of H.Its size is kn

2+ t

2.

Proof. Every edge h ∈ H intersects an edge in M in at least k − 1vertices. Clearly, if h ∈ M then it is fractionally covered by g1. If h isa mimicking edge for some mi ∈M , then by Claim 4.11 it contains pi,and therefore it is fractionally covered by g1. Finally, if h /∈M is not amimicking edge, then it must intersect at least two edges in M in k−1vertices each, and therefore it is fractionally covered by g1. �


Another useful observation is the following.

Claim 4.13. Let e be an edge in E(H) \M , such that for every 1 ≤i ≤ t, pi is not contained in e. Then there exists m ∈ M2 such that|e ∩m| = k − 1.

Proof. Assume that there exists e ∈ E(H)\M contradicting the propo-

sition. Then by Claim 4.11, e lies in E(H)\(M ∪

⋃ti=1 Fi

). Therefore,

e intersects at least two edges in M in k − 1 vertices each. If e doesnot intersect an edge in M2 in k − 1 vertices, then it must intersect atleast two edges in M1, each one in k − 1 vertices. Let {mj | j ∈ J},J ⊆ [t], be those edges in M1 intersecting e in k− 1 vertices. For everyj ∈ J choose fj ∈ Fj. We claim that the set

M ∪ {fj | j ∈ J} ∪ {e} \ {mj | j ∈ J}is a (k−1)-matching in H of size n+1, which is clearly a contradiction.Claim 4.13 will follow from the next two claims:

Claim 4.14. For every j ∈ J , |e ∩ fj| ≤ k − 2.

Indeed, both e and fj intersect mj in (k − 1) vertices, and e ∩mj 6=fj ∩ mj (since fj ∩ mj = pj and e does not contain pj). Since H isk-partite, this implies that |e ∩ fj| ≤ k − 2.

Claim 4.15. For every i, j ∈ J , |fi ∩ fj| ≤ k − 2.

suppose to the contrary that |fi ∩ fj| = k − 1. Since H is k-partiteand |mi ∩mj| ≤ k − 2, |mi ∩ fj| ≤ k − 2, and |mj ∩ fi| ≤ k − 2, wemust have that

mi = {v1, . . . , vk−2, u1, u2}, mj = {v1, . . . , vk−2, w1, w2},

fi = {v1, . . . , vk−2, u1, z}, fj = {v1, . . . , vk−2, w1, z},where vi, ui, wi, z are all distinct vertices. Moreover, we have pi ={v1, . . . , vk−2, u1} and pj = {v1, . . . , vk−2, w1}. But now, the fact that eintersects both mi and mj in k−1 vertices each implies that either e ={v1, . . . , vk−2, u1, w2} or e = {v1, . . . , vk−2, w1, u2}, and thus it containseither pi or pj, contradicting the assumption on e.

This concludes the proof of Claim 4.15 and thus also of Claim 4.13�

Claim 4.13 implies:

Claim 4.16. The function g2 :(V (H)k−1

)→ R+, in which every pi, 1 ≤

i ≤ t, and every set in(mj

k−1

), t + 1 ≤ j ≤ n, are mapped to 1, is a

(k − 1)-cover of H. Its size is kn− (k − 1)t.


We are now ready to prove Theorem 4.10. If t ≤ kn2k−1 then by Claim

4.12 there exists a fractional (k − 1)-cover of H of size kn2

+ t2≤ k2n

2k−1 .

If t > kn2k−1 , then by Claim 4.16 there exists a (k− 1)-cover of H of size

kn− (k − 1)t < k2n2k−1 .

References

[1] R. Aharoni, Ryser’s conjecture for 3-partite 3-graphs, Combinatorica 21 (2001)1–4.

[2] R.L. Brooks, On colouring the nodes of a network, Mathematical Proceedingsof the Cambridge Philosophical Society 37 (1941), 194-197.

[3] P. Erdos and R. Rado, Intersection theorems for systems of sets, Jour. LondonMathematical Society, Second Series, 35 (1)(1960), 85-90.

[4] Z. Furedi, Maximum degree and fractional matchings in uniform hypergraphs,Combinatorica 1 (1981), 155–162.

[5] P. E. Haxell, Packing and covering triangles in graphs. Discrete Math. 195(1999), 251–254.

[6] P. E. Haxell and Y.Kohayakawa, Packing and covering triangles in 3-partitegraphs. Graphs Combin. 14 (1998), 1–10.

[7] J. R. Henderson, Permutation Decompositions of (0, 1)-matricesand decomposition transversals, Ph.D. Thesis, Caltech (1971),http://thesis.library.caltech.edu/5726.

[8] M. Krivelevich, On a conjecture of Tuza about packing and covering of trian-gles, Discrete Math. 142 (1995), 281–286.

[9] L. Lovasz, On minimax theorems of combinatorics, Ph.D thesis, MatemathikaiLapok 26 (1975), 209–264 (in Hungarian).

[10] W. Stanton, Some Ramsey-type numbers and the independence ratio, Trans.Am. Math. Soc. 256 (1979).

[11] Zs. Tuza, A Conjecture, Finite and Infinite Sets, Eger, Hungary 1981, A.Hajnal, L. Lovasz, V.T. S6s (Eds.),Proc. Colloq. Math. Soc. J. Bolyai, 37,North-Holland, Amsterdam, 1984, p. 888.

Department of Mathematics, Technion, Haifa, IsraelE-mail address: [email protected]

Department of Mathematics, University of Michigan, Ann ArborE-mail address: [email protected]

http://thesis.library.caltech.edu/5726

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